Introduction

There is increasing concern with collective memory -- with how the past is remembered. This has renewed debate about how history is remembered and variously rewritten to further particular agendas (Ludmila Isurin, Collective Remembering: memory in the world and in the mind, 2017; Michelle L. Meade, et al, Collaborative Remembering: theories, research, and applications, 2017; Alin Coman, et al., Collective Memory from a Psychological Perspective, International Journal of Politics, Culture, and Society, 22, 2009).

The concern here is with the symbols through which collective memory is enabled, notably in the light of insights into the manner in which this has been achieved in civilizations of the past (Lynne Kelly, The Memory Code, 2016). The question is how to frame the quest for approaches of relevance to the present and the future, as separately argued (In Quest of Mnemonic Catalysts --
for comprehension of complex psychosocial dynamics, 2007). This follows from earlier concern with erosion of collective memory (Societal Learning and the Erosion of Collective Memory, 1980).
The exploration here exploits the metaphors through which two contrasting modes of comprehension are frequently framed -- namely those of the "head" and those of the "heart". Those of the "head" are most evident in the mode of rational argument typical of the academic, political, military and business worlds -- caricatured as "talking heads". These are a feature of the optimistic logic of ever greater integrative globalization, most readily symbolized as a sphere and commonly depicted in two dimensions as a circle. Those of the "heart" are most evident in discourse emphasizing lived experience and the associated feelings -- especially of traumatic events of the past. The coherence of these feelings, especially with respect to interpersonal relations, is widely depicted with the symbol of a heart . Those identifying with that modality are typically caricatured in their turn as "bleeding hearts" -- in contrast to those deprecated as being "hard hearted" in priding themselves on being "hard headed".

These two extremes are a challenge to each other which can be usefully caricatured as a continuing battle between the "headless hearts" and the "heartless heads". Proponents of the "head modality" deplore their lack of credibility in the eyes of those attaching far greater weight to the "heart modality". Proponents of the heart modality regret the heartlessness and impersonality characteristic of the models and policies of those of the head modality. Arguably each modality reflects a partial insight whose limitations undermine any more integrative comprehension.

Central to that comprehension might be the nature of care as it is understood within each modality -- compassion for the one, vigilance for the other. These would be matched by the nature of carelessness in each case -- a blindspot with regard to the future and negligence of systemic issues beyond those which attract immediate attention, whether from a head or heart perspective.

It is of course the case that the "heads" may well experience empathy/sympathy and antipathy/animosity for each other to a degree which may be fundamental to their relationships -- and giving rise to epic disputes (Knowledge Processes Neglected by Science: insights from the crisis of science and belief, 2012). Similarly the "hearts" may be called upon to "use their heads" -- notably in navigating romantic affinities. The question is through what richer symbolism the two modalities might be related to enable a more fruitful mode of understanding.

The epic battle between them is now epitomized by their respective approaches to the refugee crisis, most notably in Europe. The "hearts" advocate an open-armed response -- at any cost to the cultures receiving them. The "heads" express caution and increasing resistance -- expressed through populism -- setting aside their economic benefits from the sale of arms to countries engendering such displacement. Curiously both avoid the the long-term implications of unconstrained flow of desperate people from countries with ever increasing populations. Both are careless in that respect.

The reconciliation of the contrasting symbols through the dynamics associated with the cardioid offers a means of highlighting a fourfold set of processes deemed fundamental to group dynamics and psychosocial cohesion -- as exemplified by flocking behaviour and simulation of "boids", as increasingly evident on social media (Dynamically Gated Conceptual Communities, 2004). As framed by Jamie Davies (Life Unfolding: how the human body creates itself, 2014), these are: attraction (promising), repulsion (divorcing/alienation), searching/researching, and alignment (compromise),

Ring, loop and Ouroboros

An appropriate symbol of relevance to this argument -- the ring -- is curiously shared by the "heads" and the "hearts".

For the heads, it features extensively in mathematics in the form of the circle and its transformations. It implies various forms of closure in argument. It is the basis of the dynamics of the ring structure of the benzene molecule -- as a resonance hybrid -- so fundamental to organic life. In the form of the dynamics of a feedback loop, the ring is vital to the design and operation of process control systems. As a consequence, social policy is variously concerned with vicious loops (and cycles of violence) and the possibility of virtuous loops.

Beyond Earth, life without DNA is just about thinkable.... Life without feedback loops, though? I have never met any biologist who can imagine that. The helix is too well-established an icon to be deposed any time soon. And yet, a simple loop would be a much more universal symbol of how life works at all of its scales and levels. Perhaps the Ouroboros, beloved of gnostics and alchemists, has been an ideal symbol waiting in the wings for centuries: there can surely be no more evocative symbol of feedback than a snake growing by devouring its own tail.

Breaching the "circle of trust": contemporary tragedy of the "broken ring"

The widely noted breakdown of social relationships can be usefully characterized in terms of "broken ring" symbolism. This could imply that it is increasingly less feasible to "circle the wagons" -- other than in the most limited and defensive manner. The implications are of course evident in the increasing incidence of divorce, given the manner in which the ring symbolizes the marital bond. They are of course evident in the loss of credibility of ritual use of circlets of prayer beads.

The inability to engage effectively with this potential for failure is a feature of attraction to utopias. It is usefully summarized in a description of the psychosocial preoccupation of Félix Guatteri:

No matter how communal the initial struggle, sooner or later the collective will dissolved into a competition between individual desires – with one person eventually emerging as the leader, at the expense of the others. Why do collaborations always collapse into hierarchies, he asked himself? Why does the group get atomised, rather than retaining a unified voice? (Edward Thornton, Two's a Crowd, Aeon).

Such "negative" conclusions raise the question whether the current preoccupation with the "positive" detracts from a more healthy comprehension of the "negative" -- as remarkably illustrated through their role in electricity. This suggests that there is a dynamic between "positive" and "negative" which has as yet to be rendered comprehensible in psychosocial system dynamics. The beating heart, if only as an electromechanical system, offers clues in that respect.

It is in this respect that there is considerable irony to use of the heart symbol on the cover of a special issue of New Philosopher -- a symbol to which the many contributors make no reference. The editorial does however highlight the challenge of the "heartbroken".

Philosophers as "heartless heads"?
Encaging life through unintegrated formalisms
-- only too obviously characterized by "holes"

Pondering the meaning of life is not just the domain of Pythons and philosophers. Curious eight-year-olds, heartbroken teens, and expiring octogenarians alike reflect on life's meaning, if only in difficult or dull moments. And, much like the legions of thinkers before them, they can't agree on an answer. Some say it's 'love'; others 'learning how to die'; or 'flourishing'; or, one that's particularly popular among teens and philosophers, 'there is no meaning'.
(Editorial extract; emphasis added)

Whether in cybernetic or symbolic terms, it can be usefully asked "what flows" through an unbroken ring that is variously understood to be so vital. From a cybernetic perspective this can be readily seen as the information characteristic of feedback loops. In symbolic terms, as "circulation of the light", this has been explored more fundamentally by Carl Jung and Richard Wilhelm with respect to a Chinese classic, The Secret of the Golden Flower (Tai Yi Jin Hua Zong Zhi). This is discussed separately with respect to its current implications (Circulation of the Light: essential metaphor of global sustainability? 2010). Much can also be made of the hole in the ring in symbolic terms.

If the ring -- in the form of the loop or Ouroboros -- is the ultimate symbol, it clearly suffers from the disadvantage of being too abstract for comprehension in systemically grounded practice. Through its simplicity it is indeed memorable and valid for what it implies -- but at a level of abstraction which renders its significance elusive (however necessarily so). Ironically it might well be said that it is difficult to "get one's head around it". Clearly the use of the rosary and other circlets of prayer beads are a challenge to such an assertion. They offer one sense of what "flows around the ring", although seemingly less meaningfully for many than in the past.

There is clearly the sense that any elusive "ring holders" are failing to perform their "ring binding" function -- strangely echoing features of the modern myth-making of Lord of the Rings. This is especially curious in that social media like Facebook offer a sense of intertwining the modalities of the head and the heart.

As illustrated below, the arrangement also echoes that of various much-cited patterns of significance to the study of cognition and associated experience, as separately discussed (Visualization in 3D of a trinity of connotations as a cognitive pill, 2017). The relevant controversies, preumably predictable, are usefully reviewed by Maurizio Meloni (A Triangle of Thoughts: Girard, Freud, Lacan, Psychomedia, 14, 2002). That of Jacques Lacan is a version of the triangulated Oedipus complex (mother-child-father) combining Freud's theory with structural linguistics, developed from the theories of Saussure, Levi-Strauss and Jakobson (John Phillips, Lacan and Language).

The image of Ron Atkin is used to illustrate the challenge of integration beyond any binary clash using the mathematics of q-analysis (Multidimensional Man; can man live in 3-dimensional space?, 1981), as separately summarized (Comprehension: Social organization determined by incommunicability of insights). Using the geometry of a triangle, Atkin illustrates the challenge of comprehension in relation to experience "within" -- especially with regard to the perspective necessary to comprehend the geometry of the triangle as a whole -- namely the trinity. The perceptual significance of this approach is well-illustrated by visual sensitivity to colours resulting from the three primary hues (red, green and blue). These are represented on a simple triangle (below). Here the vertices (O-simplexes) represent the primary hues, the sides are twofold combinations (1-simplexes), and the combination of the three hues makes the central white (2-simplex).

As illustrated by the schematic above-right, there is a sense in which such representation is a radical form of reductionsm in which the "cognitive curvature" with which comprehensions is associated is conflated with the "cognitive linearity" of a highly questionable form of simplification, itself usefully illustrated compared with those which follow. Typically an "explanation" of each line is offered separately, notably using a curve as a form of "meta-explanation" to offer comprehensible meaning to the relations between the points of that triangle -- considered to be especially fundamental. That meta-relationship may be especially challenging as in the case of the Christian Trinity.

Especially striking is an unusial depiction of the Christian Trinity (below-left) from long before the formal recognition of the Borromean rings. Such depiction could be explored further in the light of the recent declaration by the UN Secretary-General (UN must reform to defend enlightenment values, secretary-general says, The Guardian, 10 May 2017) pointing to the three separate pillars of the UN: peace and security, human rights and sustainable development. He asked whether these could continue to be addressed separately, calling for them to be combined into a single program -- one central to any strategy of of reform.

To the extent that dynamics imply forms of curvature, anticipating discussion below of the curves of the heart symbol (and their dynamics), another approach is to use the 2D triangle as an encoding template for significance that is typically only implied by the linearity of its form -- if at all. Can the triangle -- favoured by the "head modality" -- be considered an approximation to the heart symbol -- as favoured by the "heart modlity"? The question is relevant to reframing the triadic forms presented above.

The nature of that potentially misleading approximation to the ring -- or to the heart symbol -- is evident from the image below-left. When its inversion is superimposed, as also illustrated below, it is suggestive of a need for comprehension which is primarily implicit -- and of potential significance to any interpretation of the Star of David. A speculative exploration of the dynamics inherent in that symbol in 2D is offered by its use as a template for encoding the 64 transformation conditions of the I Ching, as discussed separately -- from which the animations are reproduced (Mapping of I Ching hexagram coding onto Star of David, 2008; Star of David as reinforcing dangerous cognitive reductionism? 2017).

Reference to both the heart and depictions of the Christian Trinity also recalls the great importance attached to their association with the Sacred Heart -- one of the most widely practiced and well-known Roman Catholic devotions.

Heart symbol as articulation of ring symbol

It is of course the case that affection is widely expressed and remembered through representations of the heart symbol -- most obviously through communications worldwide on Valentine's Day. In 1962 the art historian Erwin Panofsky noted that the Valentine heart shape has been used since prehistoric times and was seen in Palaeolithic cave paintings in Spain. Note however the elimination of the hole from the heart symbol (despite its profound significance). A "hole in the heart" is then highly problematic. Curiously the symbol is unexpectedly featured in one of the main financial journals (Marilyn Yalom, The Mysterious Origins of the Enduring Heart Symbol, The Wall Street Journal, 26 January 2018).

As one of the vital organs, the heart was long identified as the center of the entire body, the seat of life, or emotion, or reason, will, intellect, purpose or the mind. The heart is an emblematic symbol in many religions, signifying truth, conscience or moral courage in many religions – the temple or throne of God in Islamic and Judeo-Christian thought; the divine centre, or atman, and the third eye of transcendent wisdom in Hinduism; the diamond of purity and essence of the Buddha; the Taoist centre of understanding

One clue to any more fruitful articulation of the ring is offered by "re-membering" the otherwise disparate and distinctive phases of various cycles -- of which feedback loops are one form. It is of course the case that the viability of the human body is dependent on the distinctive phases of the beating heart in the cardiac cycle -- of which there are a number of dynamic representations. Their symbolic significance is discussed further below. Further insights are probably to be found in the preoccupations of the HeartMath Institute, notably their program of Global Coherence Research as the "Science of Interconnectivity" and their concern with heart intelligence.

It has long been recognized that those of the "head" can be comically ignorant of "matters of the heart" -- just as those of the "heart" find the preoccupations of the "head" to be totally mysterious, if not completely alienating and irrelevant to their daily lives. Whether identified with the head or the heart, it would be appropriate to assert that most humans spend their lives in "vehicles" about whose operation they are essentially clueless -- and are happy to remain so, as with their use of an automobile or aircraft. Arguably this is true of global civilization as a whole, in the light of the arguments of John Ralston Saul (Unconscious Civilization, 1995).

The challenge can be seen otherwise in that intelligence is currently most typically associated with the "head" in contrast to emotion -- most typically associated with the heart. The work of Daniel Goleman has contributed significantly to reframing this distinction and development of insight into emotional intelligence in contrast with IQ (Emotional Intelligence: why it can matter more than IQ, 1995). Appropriately this has enabled recognition of social intelligence (Social Intelligence: the new science of human relationships, 2006). What consideration is given to either of these forms of intelligence in elaborating global strategic initiatives?

Such concerns are being taken seriously by academia with the recent establishment of a Collaborative Research Center "Affective Societies" by the German Research Foundation (DFG). In recognizing that Affect and emotion are not at the margins of societal affairs, but at the heart of civilization itself, its preoccupations are clearly of relevance to this argument:

[It] aims at developing a new understanding of societies as Affective Societies that in its essence accounts for the fundamental importance of affect and emotion for human social coexistence in the mobile and networked worlds of the 21st century. The title Affective Societies signifies the fact that societal processes are intimately linked to the complex and often antagonistic feelings of groups and individuals. Only a sound understanding of the affective dynamics of societal and communal coexistence will allow societies to utilize the cohesive forces of affect and emotion and to recognize their disruptive potential at an early stage.... They can produce new forms of communal affective bonds that transcend national and geographical boundaries, but may likewise create disintegrative tensions, frictions and conflict between social groups.

Such explicit reognition of the "heart of civilization" highlights the irony of the current status of Mesopotamia, long claimed as the historical heart (Curtis Ford, Mesopotamia: the heart of civilization, 2017). How indeed will the future appreciate the degree of systematic destruction wrecked upon the cradle of civilization in the past decades by the societies which arose from it? Engendering a "hole in the heart"?

Possibilities of interrelating ring and heart curve

Is the challenge of articulating the ring usefully to be seen as one of deconstructing the ring into its geometric elements and recombining them in ways which distinguish distinctive cognitive phases? Their distinctiveness could then be seen as a cognitive reflection of the contrasting dynamics within a viable pumping heart.

Rolling cycle: One is indeed generated dynamically by rolling a circle around another of the same radius as shown below -- although this does not give rise to the preferred form of the heart as so widely depicted.

Cardioid generated by a rolling circle on a circle with the same radius

Cardioid as envelope of a pencil of circles

Reproduced from Wikipedia

Heart pattern from fourfold configuration of circles: Geometry offers perhaps the simplest static alternative to representing a relation between a heart curve and a ring, as shown below.

Heart framed by circular geometry

Heart pattern from lauburu as a fourfold configuration of circles: A more complex method of eliciting a heart curve is indicated below in relation to the geometric construction of the Basque symbol of the lauburu.

Versions of lauburu and their superposition

Eliciting the heart pattern from geometry of lauburu
(rotated 45 degrees, with superposition of both variants)

Aesthetic dimensions of the heart?

Arguably the preference for any specific form of the heart curve as a symbol would be influenced to a relatively high degree by aesthetic proportions. This would be especially the case if that symbol is recognized as reconciling divisive perspective, as argued separately (Requisite attributes of a mandala relating concord and discord? 2016).

It is therefore intriguing to recognize that the aesthetic proportions of the golden ratio could be seen as a feature of the construction of one form of heart pattern -- as elicited through the construction of the lauburu. As depicted above, the dashed lines which are a feature of that construction are in golden ratio proportion, widely denoted by the Greek letter phi. In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. (Eric W. Weisstein, Golden Ratio, MathWorld; Keith Devlin, The Myth That Will Not Go Away, May 2007). The appeal of those proportions has been fundamental to art and architecture down the centuries and is the topic of extensive commentary. [NB: A more precise mode of construction is required to confirm the manner in which the patttern is apparently "locked", as shown above]

Rather than associating that proportion with the construction of the lauburu, the golden ratio can be used to elicit a form of the heart pattern from the juxtaposition of four circles. The process is indicated by the image on the left below where the centres of two upper circles are positioned on a central (coloured) circle. The length of the dashed line separating them is used to determine a second (longer) dashed line below in terms of the golden ratio. The extremities of that line are used as the centres of a pair of two circles of larger diameter which are then tangential to the smaller circles above. This elicits a heart pattern relating the four circles as highlighted with a line in bold. The image can of course be reversed as shown in the central image. The two images can also be superimposed as shown on the right. As discussed further below, the pairs of circles can be understood as the cross section of a torus in 3D -- the juxtaposition of the two tori forming a heart pattern in 3D.

The constructions above can be used to provide a potentially valuable "story" of the quest for systemic integration of feedback loops. This is partially illustrated by the following screen shots which form part of the more extensive animation on the right. The animation indicates a variety of assumptions and approximations to such integration -- the different curves to be understood as portions of feedback loops which variously "fail to loop", namely forms of "broken ring".

The animation suggests that the significance of the ring symbol is inherently elusive, other than through intuited implication, whereas the heart pattern is inherently more comprehensible -- despite its requisite complexity (discussed further below).

Ironically some phases in the animation suggest that the curves nested within the circle of the traditional symbol of the Tao may well prove to be a succinct integration of the circle/sphere and the heart symbol. Given the abstraction associated with that traditionl symbol, again the challenge would still remain to offer a degree of articulation to the heart symbol. Mention of this possibility anticipates later discussion below regarding speculative 3D depiction of the Double Möbius / Tao / Helix / Hydrogen Atom sketch of Erwin Schrödinger from his Notebook (see Walter Moore, Schrödinger: Life and Thought, 1992, p. 193; Doug Renselle, Schrödinger's Lissajous to Tao and Mobius Morph, Quantonics, 2015).

With respect to any such spherical reconciliation of "head" and "heart" in 3D, the point to be stressed is that "globalization" is readily to be recognized as an imagination "of the head" with which "the heart" is much challenged to identify in some way in order to give credibility to global remedial strategies in this period. The imagination "of the heart" has yet to be widely enabled in stretching to that comprehension. Some relevant arguments are indicated separately (Engendering Invagination and Gastrulation of Globalization: reconstructive insights from the sciences and the humanities, 2010).

Interpretation of the heart pattern via the coaction cardioid

Such configuration opens the possibility of exploring how the features of the heart pattern might be interpreted in more detail in ways of relevance to domains and conditions in which feedback loops are of vital to system viability.

In the cybernetic analysis of the more complex and organized systems we recognize
two distinct kinds of factors. There is the work component or components,
which we shall designate X, and the governor, or controller, which we shall
designate Y [see table below]. Of course, the governor does work too (the strategic work), and
we have simplified the relationships very greatly. There will be cases of
a system made up of sub-systems, one controlling in some respects, not in
others, and so on. Let us stay with the simpler case. Now, the processes that
characterize X may, in the interaction with Y, be accelerated or in some way
enhanced ( + ), or may be unaffected ( O ), or may be decreased ( - ). Similarly,
the processes that Y undergoes. When the possibilities are cross-tabulated,
it becomes evident that there are nine and only nine of these qualitatively
different `coactions.' [glossary]

Haskell applies this insight to a range of systems, most notably those in the natural
environment, but also in the social environment. In the case of
the different kinds of relations between animals in an ecosystem, the following
patterns then emerge -- of which 8 of the 9 are non-neutral. Note that there
are variations in the terminology of biological interaction, notably differing
from Haskell's usage [more | more].
The dynamics of each of the 8 relationships might be described as a "game",
however asymmetrical or predictable the outcome (as with the "cat-and-mouse"
game of predation).

This articulation allows distinctions to be made between forms of "being positive"
such as "symbiosis" in which there is indeed mutual enhancement. This
contrasts with one such as "predation" in which one party (the controlling
one) in a transaction prides itself on achieving a "positive" outcome
at the expense of the other -- namely "feeding off" the other. This
is distinct from the situation of "parasitism" where it is the latter
party (the subordinated one) which successfully feeds off the former.

The point
about such a table is its merit in avoiding confusion between different forms
of "being positive", some of which may be quite problematic because
their "negative" aspects are camouflaged by appearing to borrow some
of the desirable qualities of the mutuality of "symbiosis". Parasitism
and predation are not relationships of mutually beneficial mutuality. One party
effectively benefits at the expense of the other.

Haskell introduced the neologisms indicated in the above figure:

allotrophy (derived from the Greek for "feeding the other"), namely one in which the activity of the system's work is not affected while that of the controller is unaffected

allopathy (derived from the Greek for "making the other sick"), namely in which the activity of the system's work component is unaffected, but that of its controller is decreased

Similarly what might be considered most problematic is a form of (double)
"negativity", typical of "synnecrosis" in the table, in
which both parties lose energy through the interaction to the point of mutual
destruction. But again this should not be confused with other hybrid conditions
in which one or other may benefit unequally from the interaction. The table
is an indication of the possible range of interactions between positive and
negative as a form of psycho-social cybernetic system.

The table is particularly significant in that, as with environmental systems,
it is not the case that all "parasitism" or "predation"
should be eliminated from the pattern of psycho-social interactions -- however
much there is an expectation that the "Lion will lie down with the Lamb"
in an ideal form of symbiosis. There are in fact conditions under which even synnecrosis
would appear to be appropriate -- as in decay processes necessary as precursors
to regeneration. The real challenge is to ensure an appropriate systemic balance
between the various forms of interaction - and, metaphorically, to avoid."throwing
the baby out with the bathwater".

Coaction cardioid variously oriented

Upright heart pattern

Haskell's Cartesian orientation

Inverted heart pattern

Edward Haskell's insights have been very usefully (and extensively) adapted
by Timothy Wilken (The
Relationship Continuum, 2002) to an ordering of the spectrum of personal
relationships: adversity -- neutrality -- synergy. Wilken equates "synergy"
with "positive" and "adversity" with "negative"
therefore pointing to the relevance of his adaptation to the preoccupation of
this paper. Wilken's study reframes Haskell's above ordering in the following
table, where "win" equates with "positive" and "lose"
with "negative"

Relating the heart pattern to the magic square and the BaGua mirror

This exploration was triggered by the question of how to articulate the integrative ring -- the heart pattern being a possibility to which many can relate. As a potential source of insight into such articulation, two other patterns can be explored, especially to the extent that they themselves can be related to the heart pattern. They are the magic square -- long a focus of mathematical interest -- and the BaGua "mirror", fundamental to Chinese culture. As with the experience of completing a sudoku pattern or solving Rubik's Cube, both offer a particular cognitive sense of integrity. These examples offer a valuable reminder that -- whether for those of a "heart modality" or of a "head modality" -- the challenges of the heart may indeed require puzzle-solving of the highest order (as romantic relationships have always made evident). Global sustainability merits recognition in just such terms.

With respect to any relation to the heart pattern, is "sustainability" as much a matter of the "heart" as it is of the "head"? Could the 3x3 magic square, namely of the lowest non-trivial form, be explored in the light of a heart pattern?

Would a magic square offer an approach to more systemic ordering of the 8 otherwise asystemic set of Millennium Goals as exemplifying the functional operation of the "heart" of global civilization? Their disparate appreciation through functionally uncoordinated agencies could indeed be considered as exemplifying a form of "broken heart" due to lack of recognition of the feedback loops interrelating them. The irony is that this failure is as much a failure of the "head" as of the "heart" -- a form of carelessness, as indicated above. There is little systemic preoccupation with those goals by the disciplines which might otherwise be responsible for such considerations.

It was on this basis that the 8 goals were tentatively arrayed in a 3x3 pattern in the earlier exercise. The obvious question being how best to order them to suggest and enhance their meaningful interplay. They are indicated here in relation to the BaGua pattern discussed below.

3x3 Magic square(originally described
by Lo Shu)

Towards an arrangement of
Millennium Development Goals(adjusting their official numeric identifiers
around an empty centre)

As illustrated, as an exercise, the pattern of 8 MDGs could indeed be ordered in an array corresponding to the numbering of the magic square on the left. Clearly this fails because the numbering does not extend to 9. Some such ordering would be possible if the MDGs were renumbered and particular significance was attributed to the central role of 5 in the pattern. In the case of the reframing by the UN of the 8 MDGs into the 16 Sustainable Development Goals, this distinctive significance is evident with regard to the 17th SDG. Is a higher order of integrity engendered by use of such a pattern? How is such a set to be recognized as a viable whole?

BaGua mirror: The pattern of 8 symbols, with an empty centre, is highly reminiscent of the evocative BaGua arrangement of 8 symbols -- whether denoted by trigrams or idiograms. Framed as a "BaGua mirror" (as arrayed above, right), it is traditionally associated with the "magic" of feng shui, predating the association of those symbols with the I Ching. It has a particular advantage in relation to value-goals in that it has strong qualitative connotations with personal implications. Some implications of that pattern are discussed further below.

Whereas little recognition of the ring symbol as such is accorded by politics and economics, much is made of the dynamics of economic cycles. These can be recognized as frozen rings or standing waves. Conversely claims are variously made in rhetorical terms with respect to the "heart of civilization", but with little explicit sense of what this symbolism may imply.

It could be considered remarkably curious that the heart symbol is so widely depicted in static form when the relationships with which it is associated are dynamic -- and typically dramatically so. The cover illustration of the New Philosopher (indicated above) highlights the irony of a static symbol in relation to the dynamics implied by that special issue on What is the meaning of life?

Health as a verb? As implied by the cyclic emphasis, whether physical or otherwise, it would
indeed seem appropriate that "health" should be understood through
a verb -- rather than through a noun or static quality as previously argued
with respect to other values (Happiness as a verb -- en-joying as a dynamic? 2011).

The two classical health patterns above are based on use of the pentagram in Western and Eastern cultures:

Hygiea: The Pythagoreans developed an understanding of health which they associated with the form of the pentagram and the Greek goddess Hygiea (also Hygieia or Hygeia) -- the personification of health, cleanliness and sanitation, and the origin of the term "hygiene" [more]. The name of the goddess is alleged to have been used as their primary greeting (perhaps, speculatively, to be considered an origin for "hug"). The snake encircled Bowl of Hygiea has long been used as one of the symbols of pharmacy. Hygiea Internationalis is now the official journal of the International Network for the History of Public Health.

Wu Xing: This ancient Chinese mnemonic pattern in the form of a pentagram of five phases, or five elements, is widely known in East Asia. It is traditionally associated with Chinese medicine, acupuncture, feng shui, and Taoism.
The pattern is recognized as highlighting a cyclic relationship between the five constituent elements: fire, earth, metal, water and wood. It is the vectors of the "overcoming" or "controlling" cycle which are typically represented as a pentagram. Generative and destructive cycles are shown as clockwise and counter-clockwise circular motion:

Perhaps appropriately at this time, it is the Chinese insights into the Wu Xing, expressed in metaphorcial terms, which are of greater relevance in interpreting the psychosocial implications of the implicit dynamics of the heart symbol.

Pumping system: The following table is adapted from the phases identified in the Wikipedia description of that cycle.

Reversible isothermal compression of the gas at the "cold" temperature (isothermal heat rejection)

Isentropic compression of the gas (isentropic work input).

Given frequent metaphorical reference to the economic "stomach" (Ryan Detrick, Economy strong enough to stomach a trade war, Reuters, 1 March 2018), again it might be asked what equivalent insights are offered by the metaphorical "heart". Again, use is of course frequently made of the "heart of a country" or region, but with little sense of what gets "pumped" and how. A number of cities regularly claim to be "The Heart of Europe", for example. Does this reflect anything more than superficial insight for public relations rhetorical purposes?

Electrical system: It is readily forgotten that the heart is an electrical system and the heartbeat is triggered and sustained by a pattern of electrical signals, as extensively described and illustrated by Wikipedia (Electrical conduction system of the heart). This is expressed otherwise by the Heart Rhythm Society:

First, the atrium contracts. Like a pebble dropped into a pool of water, the electrical signal from the sinus node spreads through the atria. Next, the signal travels to the area that connects the atria with the ventricles, the atrioventricular node (AV node). This electrical connection is critical. Without it, the signal would never reach the ventricles, the major pumping chambers of the heart. As the signal continues and crosses to the ventricles, it passes through another bundle of special electrical tissue called the Bundle of His. The Bundle then divides into thin, wire-like structures called the right and left bundle branches which extend into the right and left ventricles. The electrical signal next travels down the bundle branches to thin electrical fibers. Lastly, these fibers send out the signal to the muscles of the ventricles, causing them to contract and pump blood into the arteries. At rest, in a normal heart, this coordinated series of electrical signals occurs about once every second, maintaining the steady, rhythmic pattern of the heart's beat and causing a normal pulse rate of 60 beats per minute. (Electrical System)

The detailed understanding of this electrical system suggests that it could offer insights into how the dynamics of the psychosocial heart are triggered and sustained -- especially in relation to the necessary feedback loops. The heart could then be usefully understood as a complex of feedback loops. Many songs are specifically entitled "heartbeat", suggesting some intuitive recognition of this process. The metaphor is also used globally (Tracking the World's Economic Heartbeat, World Policy Journal, Summer 2014; The Heartbeat of Social Media, Social Media Today, 26 January 2012; Heartbeat of Social Media, MarketingNutz).

Systemic understanding of "Cupid's arrows" as binary cyclic triggers?

Positive and negative arrows: The argument can be taken further through the reminder offered by the symbol frequently associated with depiction of the heart symbol, namely the arrow. Especially intriguing is that the original Greek myth indicates that Cupid has two arrows at his disposal -- not only the one most commonly associated with his role (the other being frequently ignored for reasons which might well be understood as predictable):

One was cast of shining gold, and with its barbed point, Cupid inflicted wounds of love. The other arrow was made of soft silver, and its tip had the power to create hate. (Cupid's Arrows)

The myth has featured widely in many contexts down the centuries -- notably through the challenge it offers to the Apollonian perspective (as an archetypal "heartless head"). The tale by Rudyard Kipling (Cupid's Arrows, 1888) is frequently cited as a strange combination of close observation of matters of love and marriage, some mild satire of the strangeness of social conventions, and an acceptance of their strangeness. His tale has featured as The Short Story of the Day (6 August 2016).

The articulation of the coaction cardioid does indeed imply "co-action" but its representation in tabular or graphical form has reinforced the sense of a static configuration of categories. Each of those 8 categories calls for the dynamic emphasis it implies, and possibly dramatically so: predation, parasitism, etc. Given their systemic role in the integrity of ecosystems, as emphasized by Haskell, much more attention could be appropriately given to their role in psychosocial systems -- perhaps usefully clarified by interpersonal relationships as transactional games:

"predation", given preoccupation with predatory males and predatory females

"parasitism", given concerns with problematic parasitical relationships between individuals

"commensalism", given the symbolic value associated with "eating at the same table"

"amensalism", given the implications of "eating at separate tables" (in separate locations)

"allolimy/alltrophy", given the situations in which a relationship is damaging to one partner (even "poisonous") but with effects to which the other is indifferent

"allopathy", given situations in which a relationship is of benefit to one but with effects to which the other is indifferent

"synnecrosis", namely a situation in which a relationship is mutually damaging

"symbiosis", namely a situation in which a relationship is mutually beneficial

The complementarity of these transactions can of course be readily challenged if it is believed possible to have a system free of the painful processes typical of (romantic) relationships -- rather than "no pain, no gain". Each transaction calls for reframing in cognitive terms to avoid the pretence that some are not required in a viable system. The challenge in this respect is that each is inherently ambiguous in its interpretation, as best illustrated by:

"predation" is the most common accusation against capitalism, whilst the latter is widely promoted as vital to economic development

"parasitism" is a common qualification of socal security entitlements, whilst the latter are widely defended as essential charity vital to the survival of the disadvantaged

"symbiosis" may well be beneficial in daily life to both partners for a time, but this contrasts with how it is imagined by both as being "for all eternity" (given the incidence of divorce)

"synnecrosis" may well be characteristic of a mutually destructive relationship (even life threatening), however it gives meaning to the sense of "until death do us part", reframed as necessary in order for both parties to "move on" (as the counterpart to conventional contractual commitments to symbiosis)

"Electrical arrows": The axes in Haskell's depiction, and its adaptation, can then be usefully explored in terms of the binary opposition between their extremes -- as metaphorical "arrows" triggering feedback cycles. The positive (attractive love) and negative (alienating hate) characteristics can then be explored with respect to those dynamics, as tentatively indicated schematically in the animation on the left below. Given the electrical interpretation above, the "arrows" can be explored as "electrical arrows", strategic dilemmas, or systemic contradictions. As argued above, the animation lends itself to comparison with the dynamics relating the conditions of the BaGua mirror, as tentativily indicated on the right below.

Coaction cardioid animation
with positive/negative "arrows"

BaGua configuration of trigrams (as an animation)

As a tentative exercise, the emphasis in the above animations is on a degree of visual comparability. Both patterns have fundamental systemic dimensions which merit far greater attention in order to improve the comparability of the two animations. Both in borrowing the use of "arrow" on the left, their color coding and directionality, and the colour coding of the BaGua pattern on the right, the argument is that further exploration is required to justify any such comparability.

Especially interesting is the interpretation of the binary distinction of "positive" versus "negative" which is handled in a far more subtle manner in the BaGua encoding. There is considerable reflection on the significance which can be associated with the various conditions in the latter pattern -- of which the encoding is suggestive. This contrasts fruitfully with that so assiduously cultivated in Western societies (Barbara Ehrenreich, Bright-sided: How Positive Thinking Is Undermining America, 2010)

Implication of inversion: The potential challenges of such a pattern language can be taken further by exploiting the significance of the inversion of the heart symbol -- recognizing that the "upright" depiction above is already a 45 degree rotation from the conventions of the Cartesian orientation that feature in Haskell's argument. As with other quadrant depictions, that of Haskell privileges the upper right quadrant as the zone of symbiosis -- thereby readily framed as an ideal whilst effectively ignoring the potentially questionable dynamics associated with the other quadrants which enable that preferred condition.

For example, if the horizontal axis is understood to be the interface between life in its material sense and that in its imagined sense, then "symbiosis" is indicative of the capacity for mutual enrichment in daily life. The condition of "synnecrosis" is then indicative of the sense in which barriers of cognitive separation are then destroyed -- as in the ideal of consummation.

If the heart symbol is inverted, "symbiosis" in the superior position can be interpreted more conventionally as the ideal of that consummation -- whereas "synnecrosis" can be understood as the kinds of painful destruction of social structures typical of what is advocated as a prerequisite of psychosocial revolution. The necessity for destruction of outmoded patterns is also recognized in the process of individuation. The challenge is to embody these various systemic conditions and perspectives into a single dynamic symbol.

Heart symbol as a standing wave pattern of feedback loops

Superimposition of lauburu and BaGua mirror: Given the manner in which the heart pattern was elicited from the construction of the lauburu above, and the suggested correspondence with the BaGua mirror, there is a case for exploring the indications they offer with respect to feedback loops from the animations below.

Toroidal dynamics: As mentioned earlier, there is a case for recognizing that the heart symbol (however inverted) can also be fruitfully understood in 3D. The upper and lower pairs of circles are then each to be recognized as cross-sections of a torus -- again recalling the ring with which is was argued that the "heart" should be reconciled or entangled in some way. Useful depictions of this (below) were elaborated in previous arguments. The animation on the left can be interpreted in terms of vaiously activated feedback loops or pathways.

Interpretation in terms of inflation-deflation: The functioning of the heart -- its pumping action as noted above -- can also be understood in terms of the phases of "filling" and "emptying" of the four ventricles. Animations which highlight the inversion of the heart symbol can then be understood as filling or emptying the upper or lower sets of circles -- then to be understood as spheres rather than cross-sections of a torus.

Contrasting animations of heart pattern inversion

Equal circle construction

Phi-based construction

The animations above are also useful to the extent that they recall the arguments of quantum physicist David Bohm with respect to the explicit and implicit order and the associated holomovement (Wholeness and the Implicate Order, 1980). In that sense, the strongly outlined circles above are indicative of an explicit "filling" or "emptying", whereas those more faintly outlined are suggestive of a corresponding -- but implicit -- "filling" or "emptying". The latter offers an interesting depiction of the much-valued aesthetics of "negative space" as well as a reflection on the necessary dynamic between "re-membering" as "filling" and "forgetting" as "emptying".

The contrast between the extremes also invites consideration in terms of the extremes of depression and elevated mood, whether or not these take pathological form as bipolar disorder. Comparison could similarly be made between extremes of optimism and pessimism, notably in the contrasts in public discourse between "hope-mongering" and "doom-mongering".

Freudian implications of the heart pattern and its inversion? Marilyn Yalom argues that the symbol of the heart reveals a tension between love as romantic and sexual on the one hand, and as religious and spiritual on the other (The Amorous Heart: an unconventional history of love, 2018). Ultimately, the heart symbol is a guide to the astonishing variety of human affections, from the erotic to the chaste and from the unrequited to the conjugal. Yalom makes the point that Medieval artisans, without the benefit of Freud, knew exactly what they were doing when they used such sexually suggestive symbols.

From a Freudian perspective, the animations above are indeed suggestive in a number of respects, whether with respect to fantasies associated with breasts, cleavage, buttocks or intercourse. Pscychologist Galdino Pranzarone is variously cited as asserting that the origin of the heart symbol was probably the shape of human female buttocks seen from the rear, and not an actual heart (Getting to the Bottom of Valentine's Day, Fox News, 16 February 2006). Thus the Greek goddess of beauty, Aphrodite, was beautiful all over, but was unique in that her buttocks were especially beautiful: Her shapely, rounded hemispheres were so appreciated by the Greeks that they built a special temple to Aphrodite Kallipygos, which literally meant, 'Goddess with the Beautiful Buttocks'.

In this light, all the environmentally oiented interpretations of Haskell's cardioid invite interpretations in experiential terms of sexual relations. The peak experience of consummation in intercourse can be readily compared with that of symbiosis, whilst the complementary synnecrosis can be compared with the "little death" (la petitie mort). The other transactions lend themselves to similar interpretation with respect to phases or possibilities in the process of intercourse.

Cognitive engagement with complexity through articulation of the heart pattern in playing cards

The geometrical approach above can be taken further in the quest for engaging symbols. The emphasis here being on symbols which clearly engage the imagination worldwide, it is in this sense that the role of the heart symbol in relation to its complements in the suite of playing cards is of particular interest. The forms of the symbols on such cards have clearly proven to be of continuing importance over centuries -- perhaps to be seen as indicative of "cognitive sustainabilty".

It is therefore intriguing to note the transformations which interrelate the four seemingly quite disparate forms: hearts, clubs, spades and diamonds. Arguably the contrasts between them have proven viable precisely because of the differences they imply.

Playing card symbolscorrespondences between their geometric construction

The symbols might then be arrayed as follows, with the image on the right anticipating further discussion of 3D.

Arrays of playing card symbols in 2D and 3D

2D array

tetrahedral 3D array

Implications of a 3D heart symbol

The earlier argument with respect to the heart symbol noted efforts to construct a heart pattern in 3D (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005). Beyond the arguments in relation to juxtaposition of two tori, of particular interest is the manner in which the patterns could take 3-dimensional form, with all that that might imply cognitively. Each of the basic patterns (heart, diamond, spade, clubs) could for example be rotated on the vertical axis (possibly when inverted). Some would then be strikingly reminiscent of the architecture of domed buildings, like mosques, the White House, or St Peters (Rome).

Heart surfaces: There are various approaches to generation of 3D heart shapes. Least relevant are those which give depth to the 2D image (Four Ways to 3D Hearts, Graphicssoft.about.com). The Bonne map projection has the globe projected onto a heart-like shape. Of greater relevance is the work of Gabriel Taubin and Eric W. Weisstein, as variously noted (Heart Surface, MathWorld; Ivars Peterson, Algebraic Hearts, ScienceNews, 6 February 2002). Recalling the comments above on the relevance to catastrophe theory, the latter notes that a heart surface has several singularities, including a sharp point at the bottom and a deep indentation at the top. Images are presented by Michael Trott of a heart surface coloured according to Gaussian curvature and direction.

Heart surfaces indicative of radical location of "significant others"?
Screen shots of interactive demonstration of 3D heart shapes based on different equations Equations for Valentines from the Wolfram Demonstrations Project, by Michael Croucher after work by Eric W. Weisstein and Michael Trott

Toroidal reconciliation of symbols of head and heart? The earlier discussion of Cognitive heart dynamics framed by two tori in 3D (2016) drew attention to Wolfgang Daeumler's comments on the presentation of Lissajous curves on the surface of the horn torus (featured above), as illustrated by the animation below. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures.

NB: Whilst potentially interesting in its own right, the otherwise annoying flicker in the animation on the right is a consequence of the simplistic method of its development. With greater effort the blue and green curves could be presented in the same frame raher than in successive frames

As noted above with respect to some phases in a 2D animation, the curves nested within the circle of the traditional symbol of the Tao may well prove to be a succinct 2D integration of the sphere and the heart symbol -- possibly as might be imagined from the aesthetic perspective of cubism. The animation above-left takes the possibility further in that some phases are indeed suggestive of a projection of the heart symbol onto the torus as a Lissajous curve -- even more apparent with the emergence of both upright and inverted heart symbols in the animation above-right (using two complementary Lissajous curves). Other phases offer a suggestive relation to a Möbius strip, as mentioned with respect to the Double Möbius / Tao / Helix / Hydrogen Atom sketch of Erwin Schrödinger from his Notebook (see Walter Moore, Schrödinger: Life and Thought, 1992, p. 193; Doug Renselle, Schrödinger's Lissajous to Tao and Mobius Morph, Quantonics, 2015).

The animation is also somewhat reminiscent of the hypersphere animations suggestive of higher-dimensional brain functioning, as presented with respect to Transforming vehicles of identity between global and toroidal forms (2016). It is also useful in challenging misleading impressions reinforced by depiction of the heart symbol as "flat". In partcular the distinctive curves of which it is composed might be better understood as helical in nature, possibly twisting around the edges of any triangle -- with all that that may imply in terms of (paradoxical) "cognitive twists" between their extremities.

Eliciting heart patterns from tetrahedral configuration of circles: A potentially significant approach can be explored by taking the configuration of upright and inverted heart patterns elaborated above (Image 1), pulling the two hearts together to form four circles (Image 2), configuring the circles in tetrahedral form (Image 3), rotating the configuration to indicate how heart patterns might variously emerge from the configuration (Image 4), highlighting the tetrahedral framework (Images 5 and 6). The approach offers a degree of reconciliation between the ring and heart symbols -- the preoccupation of this argument.

Image 1

Image 2

Image 3

Image 4

Image 5

Image 6

Images 3-6 were generated using Stella Polyhedron Navigator. This permits the tetrahedron to be variously rotated to explore how heart patterns might be perceived in the configuration (although the lighting is such that red appears as brown at some angles, as in the images above). The challenge was how best to position the images on the faces given that the two patterns, wrapped around the tetrahedron, are effectively twisted together in an unusual way with unexpected revrsals. A solution was not immediately found, suggesting that the elements of the configuration may alternate dynamically between two complementary configurations.

The necessary twist recalls the challenge with respect to two Möbius strips. In effect the reconciliation of ring and heart symbol can seemingly only be achieved at a price, namely that associated with the paradoxicl form of a Möbius strip in which the circle is twisted. Image 2 is thus also suggestive, if it is assumed that it is "not flat" but can be recognized as constituted by 2 paired Möbius strips interlinking elements of the pattern -- as illustrated by the animations below. As a visual puzzle, Image 2 then recalls the perceptual challenge of the Necker cube. The requisite complexity, in terms of Möbius strips is usefully highlighted by the following, produced for related purposes.

Music and orbifolds: Especially provocative with respect to the relationship between head and heart are the emerging insights regarding comprehension of music readily recognized as performing a bridging function intimately related to the process of remembering and triggering a heartfelt response. It is in this sense that explorations of the organization of music by the brain is especially valuable (Dmitri Tymoczko, The Geometry of Musical Chords. Science, 313, 5783, 7 July 2006, pp. 72-74; A Geometry of Music: harmony and counterpoint in the extended common practice, 2011). This is discussed separately (Engaging creatively with hyperreality through music, 2016).

Mandalas and spokes: The interplay between the ring and the heart is usefully "regulated" by what can be seen as "arrows" (whether electrical or otherwise), or as the spokes of a wheel. The latter obviously recalls the challenge of discovering or inventing the wheel. In this context, in a knowledge-based civilization variously challenged by comprehension -- whether for an individual or a group -- this may well be a case of "rediscovering the wheel" or "reinventing the wheel". The point stressed above, with regard to the tragedy of the "broken ring", suggests that it may indeed need to be rediscovered or reinvented in some way -- with the implication that this may imply the need to comprehend it anew. In cognitive terms, any such "wheel" has long been suggested by mandala-type structures. However the above argument suggests that the quest for a "three-dimensional wheel" might prove more fruitful (Concordian Mandala as a Symbolic Nexus, 2016; Use of Concordian Mandala for a 3D ordering of value polarities, 2016; Speculation on Potential Symbolic Relevance of the Concordian Mandala, 2016 ).

Rotation of magnetic fields: With respect to the functional relationship between spokes and arrows in interrelating ring and heart, of particular interest is the well-studied manner in which spokes are alternatively under compression or tension as a consequence of parts of the wheel being in contact with the ground and then having to bear a greater proportion of the load. A richer metaphor is offered by the highly innovative work by Nikola Tesla on the relation between positive and negative in electromechanical systems, as discussed separately with animations (Potential implications of alternation and rotation in psychosocial fields, 2014).

Ring, heart and ouroboros: Part of the challenge of this exercise, as implied above, is to indicate interesting relationships between the ring, the heart and the ouroboros -- for the reasons mentioned. One such possibility is presented below-left, further to animations presented previously (Experimental animations in 3D of the ouroboros pattern, 2017). It follows from an earlier demonstration of the possibility of "stacking" 3D heart patterns using horn tori, as shown below-centre (Cognitive heart dynamics framed by two tori in 3D, 2016). That stacking is reminiscent of the architecture of stepped temples. With greater effort, the 2D animation on the left could be presented in 3D using the toroidal elements from the centre. Whilst the direction of rotation of the ouroboros is provocatively consistent with the traditional presentation of the "serpent eating its tail", it can of course be asked what the reverse rotation would imply -- especially in relation to the symbolism of the heart.

The sequence of images in the animation on the right could of course be presented in a circle -- as an Ouroboros -- as with the animation on the left.

A case can be made that patterns suggestive of the mysterious nature of the relation between head and heart can be variously found, if only speculatively. Contrasting examples are presented below. The clothoid (or Euler spiral) is for example valuable in clarifying the transition between linearity and curvature. Shown with a reflection this is potentially significant to communication more generally (Reframing communication relationships, 2012; Clothoid as a psychosocial transition curve: from linear to circular, 2012). The geometry of the heart pattern may be further transformed, as shown, in order to engender a ring (proportioned again by phi). Given their role in affairs of the heart, the human ovaries offer a strange resemblance to the heart pattern, especially given the powerful circular attractor of the vagina.

In a global civilization readily characterized as faced with fundamental communication problems, it is appropriate to explore resources creatively developed by those responding to the implications of systemic nonsense -- as discussed separately (Nonsense commensurate with dysfunctionality (2014). Reference was made there to one of the widely known "nonsense poems" by Edward Lear (The Owl and the Pussy-Cat, 1871). In relation to this argment, this can in fact be read as a relatively unique description of the possibility of a distiinctly improbable "marriage" between the head modality and the heart modality -- with a prescient environmental sensitivity. Their marriage is sealed with a ring acquired with an appropriate measure of "nonsense" from any conventional perspective.

Aside from more obvious sexual connotations, one remarkable no-nonsense commentary on the possible hidden significance of the poem is offered by David Cowles (Owl and Pussycat, Aletheia, 26 March 2014). With respect to the ring he wonders:

Could it be that the ring was actually a Möbius strip, a one sided, 2 dimensional object? A Möbius strip is 'non-orientable', a topology in which there is no fixed beginning or end and no fixed orientation. Many eschatological cosmologies, for example Dante's in the Divine Comedy, include a non-orientable topology. A Möbius strip has the unique property that allows you to travel around it continuously, always coming back to the starting point; however, each time you return to the starting point, your orientation is reversed.

Further insight is offered to the effect that:

Three dimensional objects that contain a Möbius strip are called "Klein Bottles". In our world of 3 spatial dimensions, Klein Bottles are not physically realizable, but if they were, they could hold no liquid. Why? Because like all non-orientable objects, they are one sided and therefore have no inside or outside. In this admittedly speculative interpretation, the "runcible spoon" [a feature of the marriage celebration] is the three dimensional correlate of the Piggy-wig's two dimensional ring. It suggests that there is a consistently non-orientable topology in the eschatological kingdom, "the land where the Bong-tree grows" [the place where the marriage proved possible].

The mysterious interplay between head and heart modalities, in a period in which the "circle of trust" has been breached, is curiously symbolized by the widespread use of so-called awareness ribbons (List of awareness of ribbons).

Symbol of mourning

The form of the many such ribbons echoes both that of a broken Möbius strip and an inversion of the heart symbol -- with ring -- as variously indicated above.

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know it for the first time.
(T. S. Eliot, Little Gidding, 1942).

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Voices of Collective Remembering. Cambridge University Press, 2002

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